An attempt by Senate Democrats to pass a higher tax on millionaires failed Monday night, but the legislative defeat may prove to be a political gift to Democrats eager to counter Scott Brown’s image as a bipartisan everyman.

Brown voted against the tax on millionaires, arguing that the government should not be imposing new taxes until it cuts wasteful spending.

President Obama has been leading the charge for the so-called Buffett Rule, arguing that the wealthiest generally pay lower effective tax rates than the middle class and that the situation needs to change.

The issue, pitched as a fairness argument, has received support in polls. A Gallup poll taken last week showed six in 10 voters, including independents, favor requiring those earning more than $1 million to pay a 30 percent tax rate, as the bill requires. A CNN poll released Monday put the number at seven in 10 voters.

Massachusetts Democrats are already using Brown’s opposition to the Buffett rule as a way of arguing that, on economic issues, he is closer to the national Republican Party than his image suggests.

Brown has touted two recent appearances with Obama at the White House to sign bills he cosponsored as evidence that he is a bipartisan problem-solver. He has also crossed the aisle on several prominent issues, including 2010’s Wall Street overhaul and the repeal of the military’s ban on gays and lesbians serving openly.

Monday night’s Buffett rule vote, which blocked consideration of the bill in a 51-45 tally, was timed to coincide with Tuesday’s IRS filing deadline, when many Americans are focused on their personal tax burden. Republicans prevented the measure from receiving the 60 votes necessary to open debate. All Republicans but Senator Susan Collins of Maine voted against it. All Democrats except for Mark Pryor of Arkansas voted for it.

In a statement Monday night, the White House criticized the vote as evidence of Senate Republicans “choosing once again to protect tax breaks for the wealthiest few Americans at the expense of the middle class.’’

But Brown said the higher tax would hurt small businesses and would do very little to close the budget deficit.

“The Buffett rule is a political stunt: It will raise in one year only enough revenue to pay for less than a day of federal spending,’’ he said. “It doesn’t create a single new job or cut the national debt.’’

The Democratic push for the vote was aimed primarily at presumed Republican nominee Mitt Romney, whose personal wealth has been a frequent target of political opponents in both parties.

But Democrats and outside groups have been sending out press releases on the subject for a week, criticizing Brown for his opposition.

Republicans have labeled the vote divisive political theater, and Brown did not wait to respond to Democratic attacks. He spent last week highlighting his opposition to all tax hikes, as well as a proposal to collect unpaid back taxes from federal employees. He also wrote an opinion piece in Monday’s Boston Herald, charging that “Washington politicians are simply addicted to spending other people’s money.’’

“Make no mistake, give Washington an extra dime, and they will spend a dollar,’’ he wrote.

His campaign also released a memo to reporters Friday, highlighting what it called “Elizabeth Warren’s Support For Job-Killing, Across-The-Board Tax Hikes.’’

The memo highlighted Warren’s support for President Obama’s health law, which the campaign said would raise taxes by $500 million, as well as her opposition to extending the Bush tax cuts for all Americans, which she said she opposed because it included all income levels.

Matt Canter, spokesman for the Democratic Senatorial Campaign Committee, linked Brown’s vote against the measure to Romney, who polls show is unpopular in Massachusetts.

“Scott Brown says we need to cut Medicare benefits and Social Security, but in this vote today he is refusing to ask the Mitt Romneys of the world to pay their fair share,’’ Canter wrote in an e-mail.

“I’m not sure one can overestimate the impact of an issue that so clearly shows Warren on the side of the middle class and Brown on the side of billionaires.’’

Democrats point to a string of votes Brown has taken against raising taxes on the wealthy and point out that he has signed Grover Norquist’s pledge against raising taxes.

“It’s an imaging, positioning issue,’’ said Andrew Smith, a nonpartisan pollster who heads the University of New Hampshire Survey Center.

Smith said he believes it will be a more effective strategy against Romney, who is very wealthy, than against Brown, who earned a $700,000 advance for his memoir, but whose wealth does not come close to Romney’s.

“It’s much more difficult to sustain it when you don’t have a person who looks as if he would personally benefit,’’ Smith said. “All [Brown] has to do is get out the old beat-up truck and the barn coat.’’

Though the Buffett rule is a national Democratic message, it fits particularly well with Elizabeth Warren’s rhetoric that the wealthy and influential have built-in economic advantages.

The viral video of her, shot in August, that helped cement grass-roots enthusiasm around her campaign hit squarely on the topic that the rich have a responsibility to pay society back.

But Warren, who owns stock and other investments worth more than $3 million, would not say whether she voluntarily pays a higher state income tax, which the Massachusetts form allows.

“The Buffett rule is about making sure that millionaires and billionaires pay their fair share, it’s not about funding government through voluntary contributions,’’ her campaign said in an e-mail.

“The question is whether the wealthiest citizens should somehow be excused from these obligations when everyone else is not.’’

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